In tomographic imaging, penetrating waves are used to gather projection data from an object under study from multiple directions, e.g. acquired along a range of angles. Images of the object may then be reconstructed from these projections. Different tomographic imaging modalities exist for different types of penetrating waves, for example, computed tomography generally refers to X-ray tomography, single-photon emission computed tomography and positron emission tomography refer to gamma ray tomography, magnetic resonance imaging uses radiofrequent waves, and other modalities exist for visible light, electron waves and ultrasound.
In the case of computed tomography, internal structure of a patient or object may be examined non-invasively. This typically involves the collection of projection data using a detector which performs measurements relating to x-ray beams cast through the patient or object from various angles by a moving x-ray source. This allows to calculate the distribution of beam attenuation properties inside the patient or object, for example in the single plane of rotation of the moving x-ray source, or even such a distribution in a 3D volume, for example by combining a source rotating in a plane with a translation of the object of patient in a direction perpendicular to this plane.
The reconstruction of a spatial representation, e.g. a planar image or a 3D image volume, from such projection data, may be achieved by using algorithms known in the art. The Inverse Radon Transform may provide a closed-form inversion formula for this reconstruction problem, provided that projections are available for all angles. Although this assumption is clearly not satisfied in practice, an accurate reconstruction can be computed if a large number of projections are available, for a full angular range, by using the well-known Filtered Backprojection algorithm.
Most reconstruction algorithms can be subdivided in two classes: analytical reconstruction techniques, e.g. variants of filtered backprojection (FBP), and iterative algebraic methods, such as Algebraic Reconstruction Technique (ART), Simultaneous Algebraic Reconstruction Technique (SART) or Simultaneous Iterative Reconstruction Technique (SIRT). Furthermore, hybrid reconstruction methods, which may combine both types of reconstruction algorithms, are also known in the art.
Tomographic reconstruction using algebraic reconstruction algorithms does not depend on filters, and therefore circumvents difficulties in determining suitable filters for filtered backprojection in complex imaging geometries. These algorithms may typically involve iteratively adjusting an image to minimize a difference metric between simulated projection data for this image and the measured projection data. Algebraic reconstruction methods may be considered more flexible in dealing with limited data problems and noise compared to filtered backprojection. They may furthermore allow for incorporation of certain types of prior knowledge, e.g. constraints such as non-negativity of the reconstructed image, by adjusting the image between subsequent iterations. Unfortunately, the iterative nature of these methods renders them computationally more intensive.
In several applications of tomography, only few projections can be acquired. Such reconstruction problems are known as limited-data problems. In electron tomography, for example, the electron beam may damage the sample, limiting the number of projections that can be acquired. In industrial tomography for quality assurance, cost considerations impose limitations on the duration of a scan, and thereby on the number of projections.
Applying classical reconstruction algorithms such as Filtered Backprojection to limited-data problems often results in inferior reconstruction quality. Several approaches have been proposed to overcome these problems, by incorporating various forms of prior knowledge about the object in the reconstruction algorithm. Recently, advances in the field of Compressed Sensing have demonstrated high potential in obtaining a reduction of the number of required projection images by exploiting sparsity of the image with respect to a certain set of basis functions. Following a different approach, the field of discrete tomography focuses on the reconstruction of images that consist of a small, discrete set of grey values. By exploiting the knowledge of these grey values in the reconstruction algorithm, it is often possible to compute accurate reconstructions from far fewer projections than required by classical “continuous” tomography algorithms.
When reconstructing an image from a small set of projections, for example less than 20 projections, the particular set of projection angles can have a large influence on the quality of the reconstruction. The choice of the projection angles can have a crucial influence on the reconstruction quality in binary tomography, i.e., discrete tomography based on just two grey levels. Algorithms to identify optimal projection angles based on a blueprint image, which is known to be similar to the scanned object, are known in the art, e.g. such as disclosed in L. Varga, P. Balázs, A. Nagy, “Projection Selection Algorithms for Discrete Tomography,” Lecture Notes in Computer Science, 6474:390-401 (2010).